Advertisement

Acta Mathematica Hungarica

, Volume 159, Issue 2, pp 520–536 | Cite as

Light dual multinets of order six in the projective plane

  • N. Bogya
  • G. P. NagyEmail author
Article

Abstract

Embedded multinets are line arrangements of the projective plane with a rich combinatorial structure. In this paper, we first classify all abstract light dual multinets of order 6 which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.

Key words and phrases

multinet projective embedding dual 3-net point-line incidence 

Mathematics Subject Classification

05B30 13P15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barlotti, A., Strambach, K.: The geometry of binary systems. Adv. in Math. 49, 1–105 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Bartz, Multinets in \(\mathbb P\it ^2\) and \(\mathbb P\it ^3\), Ph. D. thesis, University of Oregon (2013)Google Scholar
  3. 3.
    J. Bartz and S. Yuzvinsky, Multinets in \(\mathbb P\it ^2\), in: Bridging algebra, geometry, and topology, Springer Proc. Math. Stat., 96, Springer (Cham, 2014), pp. 21–35Google Scholar
  4. 4.
    T. Beth, D. Jungnickel and H. Lenz, Design Theory, Volume I., Second Ed., Cambridge University Press (Cambridge, 1999)Google Scholar
  5. 5.
    M. Costantini and W. de Graaf, Singular, The GAP interface to Singular, Version 12.04.28 (2012), (GAP package), http://www.gap-system.org/HostedGapPackages/singular/.
  6. 6.
    W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2016).
  7. 7.
    W. Decker, S. Laplagne, G. Pfister and H. Schönemann, primdec.lib. A Singular 4-1-0 library for computing the primary decomposition and radical of ideals (2016)Google Scholar
  8. 8.
    Falk, M., Yuzvinsky, S.: Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143, 1069–1088 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    GAP – Groups, Algorithms, and Programming, Version 4.9.2, The GAP Group (2018), https://www.gap-system.org.
  10. 10.
    D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Version 1.12., https://faculty.math.illinois.edu/Macaulay2/ (2018).
  11. 11.
    Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra, Second, extended edition, with contributions by Olaf Bachmann. Christoph Lossen and Hans Schönemann, Springer (Berlin (2008)zbMATHGoogle Scholar
  12. 12.
    Keedwell, A.D., Dénes, J.: Latin Squares and their Applications. Second edition, Elsevier/North-Holland (Amsterdam (2015)zbMATHGoogle Scholar
  13. 13.
    Korchmáros, G., Nagy, G.P.: Group-labeled light dual multinets in the projective plane. Discrete Math. 341, 2121–2130 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Korchmáros, G., Nagy, G.P., Pace, N.: 3-nets realizing a group in a projective plane. J. Algebraic Comb. 39, 939–966 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Korchmáros, G., Nagy, G.P., Pace, N.: \(k\)-nets embedded in a projective plane over a field. Combinatorica 35, 63–74 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Miguel, Á., Buzunáriz, M.: A description of the resonance variety of a line combinatorics via combinatorial pencils. Graphs Combin. 25, 469–488 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. P. Nagy and N. Pace, Some computational results on small 3-nets embedded in a projective plane over a field, J. Comb. Theory, Ser. A, 120 (2013), 1632–1641MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pereira, J.V., Yuzvinsky, S.: Completely reducible hypersurfaces in a pencil. Adv. Math. 219, 672–688 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    SageMath, the Sage Mathematics Software System (Version 8.2), The Sage Developers, http://www.sagemath.org (2018).
  20. 20.
    L. H. Soicher, DESIGN, The Design Package for GAP, Version 1.6 (2011), (Refereed GAP package), http://www.designtheory.org/software/gap_design/.
  21. 21.
    J. Stipins, Old and new examples of \(k\)-nets in \(\mathbb P\it ^2\), arXiv:1104.4439v1.
  22. 22.
    Urzúa, G.: On line arrangements with applications to 3-nets. Adv. Geom. 10, 287–310 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yuzvinsky, S.: Realization of finite abelian groups by nets in \(\mathbb{P}^2\). Compos. Math. 140, 1614–1624 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yuzvinsky, S.: A new bound on the number of special fibers in a pencil of curves. Proc. Amer. Math. Soc. 137, 1641–1648 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Bolyai Institute, University of SzegedSzegedHungary
  2. 2.Department of AlgebraBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations